Dialogue Categories and Frobenius Monoids

About ten years ago, Brian Day and Ross Street discovered a beautiful and unexpected connection between the notion of ∗-autonomous category in proof theory and the notion of Frobenius algebra in mathematical physics. The purpose of the present paper is to clarify the logical content of this connecti...

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Published inComputation, Logic, Games, and Quantum Foundations - the Many Facets of Samson Abramsky Vol. 7860; pp. 197 - 224
Main Author Melliès, Paul-André
Format Book Chapter
LanguageEnglish
Published Germany Springer Berlin / Heidelberg 2013
Springer Berlin Heidelberg
SeriesLecture Notes in Computer Science
Subjects
Frobenius Algebra
Monoid Structure
Monoidal Category
Symmetric Monoidal Category
Tensorial Logic
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Abstract About ten years ago, Brian Day and Ross Street discovered a beautiful and unexpected connection between the notion of ∗-autonomous category in proof theory and the notion of Frobenius algebra in mathematical physics. The purpose of the present paper is to clarify the logical content of this connection by formulating a two-sided presentation of Frobenius algebras. The presentation is inspired by the idea that every logical dispute has two sides consisting of a Prover and of a Denier. This dialogical point of view leads us to a correspondence between dialogue categories and Frobenius pseudomonoids. The correspondence with dialogue categories refines Day and Street’s correspondence with ∗-autonomous categories in the same way as tensorial logic refines linear logic.
AbstractList About ten years ago, Brian Day and Ross Street discovered a beautiful and unexpected connection between the notion of ∗-autonomous category in proof theory and the notion of Frobenius algebra in mathematical physics. The purpose of the present paper is to clarify the logical content of this connection by formulating a two-sided presentation of Frobenius algebras. The presentation is inspired by the idea that every logical dispute has two sides consisting of a Prover and of a Denier. This dialogical point of view leads us to a correspondence between dialogue categories and Frobenius pseudomonoids. The correspondence with dialogue categories refines Day and Street’s correspondence with ∗-autonomous categories in the same way as tensorial logic refines linear logic.
Author Melliès, Paul-André
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Snippet About ten years ago, Brian Day and Ross Street discovered a beautiful and unexpected connection between the notion of ∗-autonomous category in proof theory and...
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proquest
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StartPage 197
SubjectTerms Frobenius Algebra
Monoid Structure
Monoidal Category
Symmetric Monoidal Category
Tensorial Logic
Title Dialogue Categories and Frobenius Monoids
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